Convex Hull 20121023 Convex vs Concave A polygon

Convex vs. Concave • A polygon P is convex if for every pair of

The convex hull problem concave polygon: convex polygon: • The convex hull of a

Graham’s Scan • Start at point guaranteed to be on the hull. (the point

Graham’s Runtime • Graham’s scan is O(n log n) due to initial sort of

Convex Hull by Divide-and-Conquer • First, sort all points by their x coordinate. –

Lower Tangent Example • Initially, T=(4, 7) is only a lower tangent for A.

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Convex Hull 2012/10/23

Convex vs. Concave • A polygon P is convex if for every pair of points x and y in P, the line xy is also in P; otherwise, it is called concave. P x y concave P x y convex

The convex hull problem concave polygon: convex polygon: • The convex hull of a set of planar points is the smallest convex polygon containing all of the points. 3

Graham’s Scan • Start at point guaranteed to be on the hull. (the point with the minimum y value) • Sort remaining points by polar angles of vertices relative to the first point. • Go through sorted points, keeping vertices of points that have left turns and dropping points that have right turns.

Graham’s Scan

Graham’s Runtime • Graham’s scan is O(n log n) due to initial sort of angles.

n. A more detailed algorithm

Convex Hull by Divide-and-Conquer • First, sort all points by their x coordinate. – ( O(n log n) time) • Then divide and conquer: – Find the convex hull of the left half of points. – Find the convex hull of the right half of points. – Merge the two hulls into one.

5 Hull of S by exhaustive search;

Lower Tangent Example • Initially, T=(4, 7) is only a lower tangent for A. The A loop does not execute, but the B loop increments b to 11. • But now T=(4, 11) is no longer a lower tangent for A, so the A loop decrements a to 0. • T=(0, 11) is not a lower tangent for B, so b is incremented to 12. • T=(0, 12) is a lower tangent for both A and B, and T is returned.

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